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The Great One During his 20 seasons in the NHL, Wayne Gretzky scored 50% more points than anyone who ever played professional hockey. He accomplished this amazing f...

Question

The Great One During his 20 seasons in the NHL, Wayne Gretzky scored 50% more points than anyone who ever played professional hockey. He accomplished this amazing feat while playing in 280 fewer games than Gordie Howe, the previous record holder. Here are the number of games Gretzky played during each season:79, 80, 80, 80, 74, 80, 80, 79, 64, 78, 73, 78, 74, 45, 81, 48,80, 82, 82, 70a) Create a stem-and-leaf display for these data, using split stems.b) Describe the shape of the distribution.c)

The Great One During his 20 seasons in the NHL, Wayne Gretzky scored 50% more points than anyone who ever played professional hockey. He accomplished this amazing feat while playing in 280 fewer games than Gordie Howe, the previous record holder. Here are the number of games Gretzky played during each season: 79, 80, 80, 80, 74, 80, 80, 79, 64, 78, 73, 78, 74, 45, 81, 48, 80, 82, 82, 70 a) Create a stem-and-leaf display for these data, using split stems. b) Describe the shape of the distribution. c) Describe the center and spread of this distribution. d) What unusual feature do you see? What might explain this?



Answers

The Great One During his 20 seasons in the NHL, Wayne Gretzky scored 50% more points than anyone who ever played professional hockey. He accomplished this amazing feat while playing in 280 fewer games than Gordie Howe, the previous record holder. Here are the number of games Gretzky played during each season:
79, 80, 80, 80, 74, 80, 80, 79, 64, 78, 73, 78, 74, 45, 81, 48,
80, 82, 82, 70
a) Create a stem-and-leaf display for these data, using split stems.
b) Describe the shape of the distribution.
c) Describe the center and spread of this distribution.
d) What unusual feature do you see? What might explain this?

Okay, we know Hank Aaron hit 40 more more. 41 more home runs and Babe Ruth. So let's say Exes, Babe Ruths home runs. So if I add 41 2 that I got Hank Aaron's home runs together. Hope so. If I had both of these together, they hit 1469 home runs. How many did Babe Ruth it? Or, in other words, what's that? Well, not too tough. Extra sexes to X plus 41 equals 1469. Two steps. Let's subtract 41 from both sides. So that goes away. 1469 minus 41 is 1000 428. Okay, that is equal to two acts. It's the last step is to divide by two or multiply by half. However you would like to think about that. So if I take that number and divide about you, I get that Babe Ruth hit 714 home runs, which I wouldn't be surprised if these numbers are actually true, because I know Hank Aaron actually held the home wrecker run record before Barry Bonds so feel free to check of these numbers are exactly right. Historically, or if you plug it in for X, if it does indeed work.

Okay for this problem. What we're going to do is we're comparing some home run totals. So what we want to do is you want to compare, um, two different Z scores for two different baseball players to see he was a more impressive batter, more present, home run total. And what we're doing to compare Aziz scores is we know the Z score formula is what we have minus the mean of what we're given, divided by the standard deviation of those. So what we're gonna do is we're gonna go over here to does nose and look at the 1961 totals which said that we have 61 home runs that year Roger Maris in 61 home runs. And for that year, 18.8 was the average number of home runs. And then there's use for just divide that by the standard deviation for that year, which happened to be 13.37 from our table book. That gives us a Z score of 3.15 about, and we go down here to the same thing for Mark McGwire. So what we do is we're going to take his home run total, which he had 70 home run totals that year. There was 20.7 of the average for that year and the typical very typical variation or standard deviation for that year. Waas, 20.74 Now we have twosies course compare So we do a little versus here. So in the end, we know that in 1961 I, uh, one baseball player had a Z score of 3.15 We're gonna compare that to another baseball players home run score for that year, and he had 3.86 We're out of 3.7. That happened in 1998 Mark McGwire. So the answer the question was, the 1998 was the more impressive home run total.

Okay, This problem is a follow up. Teoh. Uh, the wing Gretzky hockey data from before. So, uh, happened to do a video for that? The Vega. So for me, that's family graph of this data. So I'm just gonna paste this into what we see on our screen. Some we kind of see the representative of the data in the stem elite raft for how many goals Wayne Gretzky scored. And this question asks us to talk about, Would you use a mean median? Are mean to describe the center of this distribution? Well, we kind of look amazing. What We see some skew in this data. So if we really want to repeat, the purpose of the center is to represent typically what Wayne Gretzky does in a typical year. And so because of that, we're say, the median is more representative, so the median is more representative of the me and troubles read, Ian is a better representation of the mean and the reason why isn't wonder picture here. So the center of the graph is kind of really over here to talk about it. So the medium's better representation of the center looks like it's the seventies here. So you kind of see most years of true center of the day thinking about the center that's came to happen down here, where my cursor is right now. Um, if you did the me and so the why behind that is if you did the mean you'd want Teoh, the meat is gonna be affected over over. Affected by these low values, it's 45 in this 48. Especially so the mean It's like my students like them Teoh you some words that makes some sense here. So let's mathematical, But the meeting is dragged down. So if you talk about best the meetings dragged down by those low values. So a federal presentation of center is a median in three. Add up 45 48 we all love when the stats last How to calculate the mean make that lower. It's gonna be part C. Answer. Someone will say that for a second here. So, um, question being will ask this Actually find the median? Well, I kind of had my summary statistics show on the screen here anyways, but if we were going to shore working and look for a meeting of this. So we need to do is his 20 goals. I'm gonna say 123456789 10. So the media is really between these numbers since we have an even number of numbers. So if the average of 79 79 well, the average of 79 79 is a seven night, so that's the middle number of the data set. And again had argued that is the best, most representative number from there and then asks us to find the without finding the mean is gonna be to the left or the right And why, um, what we talk about this little bit to something mean I mean, is going to be three questions that actually find lonely me. Would you expect that to be higher or lower than the media so mean is going to be lower than the medium so again, A little bit, you know, because you see the mean calculated about our patients, but without calculating it, Uh, visually, you can think that if you add about these numbers up to get the mean, you take the some of the numbers divided by the number of numbers. That's what the meeting ISS. So just for the understanding part, if I add these numbers and started calculate the some of the numbers, they're gonna be falsely lower. He's in terms of representations, meal, lower number. So that means the meaning is gonna be lowers and touching some before, since we can see it. But the meaning will be lower without doing the full of calculations. And it's generally because of the two, two or three low values you see some. So a nice little analysis of the measures of center based on some some hockey data that we looked at in a previous problem, it all makes sense.

For this question were given a preliminary regression analysis output that a coach performed on a sample of 293 players from the NHL. And he was seeking our relationship between shots taken and goals scored. And for this question, we're asked if the necessary assumptions for then your inference are met for this problem. And so there are four main assumptions. So the 1st 1 is the linearity assumption. So the data actually has to be linearly related in order to in order for it to be valid to do, Ah, linear regression. And one thing we can do is look at the first plots that's provided in the question, which is a scatter plot of the shots taken versus the goal scored. And we want to see if that forms of a fairly straight line. So if you look at this scatter plot, it definitely looks like a linear relationship. Like you can imagine a line passing through there. The data is formed in a linear fashion, from the bottom left towards the top, right? So that's that's speaks to the linear assumption. So we can say that that has been satisfied, and the independence assumption so we'd want to know that it's a random sample of players from the NHL. We're not actually told that in the question, but that would be good to know. The other thing we can do is look at the residuals plot, which is the second plot provided in the question. And here we want to see if the data if thieves scatter, is a random scatter of data. We don't want to see any clumping or patterns or trends in the data. So looking at the scatter plot of the residuals, it does look very good. It looks quite random, and I would say that the Independence assumption is there for a minute. And now the 3rd 1 is the equal variance assumption. So again, we use the residual scatter plot for this, and what we want to see is we want to see steady variants as we move from left to right in the scatter plot. So the variances is how far the data scatter vertically for a given shots. So for a given point on the X axis, how far to the data scatter? We want that to be about the same as we move from left to right, so we don't want to see a fanning out effect or a fanning in effect as you move from left to right. You just want to see a kind of kind of like a horizontal column of random data. And I would say that the scatter plot for residuals in this question is very good. In that regard, I would say that the equal variance assumption is met and forth. We want the errors where the residuals to be normally distributed. And so the third plot given in the question is a normal probability plot. And for normally distributed data, a normal probability of plot of that data would be a straight line from the bottom left to the top, right? So we want to assess how straight this Linus in this line is quite straight. There's just a few very minor bends. This is a really good example of data that is normally distributed, so we can say that the four assumptions necessary for inference have been met.

Okay, so for this problem, we're talking about Kobe Bryant and Hiss, Um, and some of his scoring points. So what we're gonna do is we're gonna call X is being the foul shots. Why? Being the two point shots or his regular filled shots and then z being the three pointer shots? So it talks about that. The total shots that he made, um, where it was 46 baskets. So X plus y plus z is gonna equal 46. And then it says that, um, he made three times as many three pointers as he did filled golds or the two point shots. So it's gonna be why minus three Z equals zero or three equals e uh, three Z. And then it says he, um that he scored a total of 81 points. So I want to dio, um X because the foul shots or one point course. The two pointers air, two points and three pointers or three points for a total of 81 points. So now what I can do is I can set up a matrix and so I'm gonna set this up. So I got 10 one 112 one negative 33 Remember, I'm just doing the coefficients. And then the totals. Air 46 0 and 81. And so, after utilizing a calculator, we've matrix calculator. We find out that we have 18 foul shots, 21 2 pointers and 73 pointers.


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